#132859

order	512
nilpotency class	4
isoclinism	isoclinic class of #132521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#2812	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

539 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,4,14)(3,11,12,7,13,8,15,16)(17,18)(19,27,28,44)(20,25,31,41)(21,33,35,24)(22,26)(23,29)(30,42)(32,39)(34,40)(36,47,38,48)(37,45)(43,46);(1,3)(2,7)(4,15)(5,12)(6,13)(8,9)(10,16)(11,14)(17,19,22,28)(18,23,26,37)(20,29,31,45)(21,30,35,46)(24,38,33,36)(25,27,41,44)(32,47,34,48)(39,43,40,42);(1,3)(2,8)(4,12)(5,15)(6,13)(7,9)(10,11)(14,16)(18,24)(19,28)(20,31)(23,36)(25,39)(26,33)(27,42)(30,46)(32,34)(37,38)(40,41)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40055);